Algorithm 1 Procedure of Secret Key Generation

1:functionKeyGen($\lambda$) 2:    Select random nonzero values from $\mathcal{K}$ to generate 8             diagonal matrices $\{{\mathit{P}}_{\mathbf{1}},{\mathit{P}}_{\mathbf{2}},{\mathit{P}}_{\mathbf{3}},{\mathit{P}}_{\mathbf{4}},{\mathit{Q}}_{\mathbf{1}},{\mathit{Q}}_{\mathbf{2}},$             ${\mathit{Q}}_{\mathbf{3}},{\mathit{Q}}_{\mathbf{4}}\}$ 3:    Pick 8 random parameters satisfying             $n\le n_1,n_2,n_3,n_4,m_1,m_2,m_3,m_4\le 2n$ 4:    for $i=1\to 4$ do 5:        $t_i\leftarrow {(-1)}^{n_i+m_i}{\prod }_{j=1}^n{P}_i(j,j)Q_i(j,j)$ 6:    end for 7:    for $i=1\to 4$ do 8:        for $j=1\to n_i$ do 9:           Randomly select two rows of ${\mathit{P}}_{\mathit{i}}$ and exchange them. 10:        end for 11:        for $j=1\to m_i$ do 12:           Randomly select two columns of ${\mathit{Q}}_{\mathit{i}}$ and exchange them. 13:        end for 14:    end for 15:    return $S{K}_1\leftarrow \{{\mathit{P}}_{\mathbf{1}},{\mathit{P}}_{\mathbf{2}},{\mathit{P}}_{\mathbf{3}},{\mathit{P}}_{\mathbf{4}},{\mathit{Q}}_{\mathbf{1}},{\mathit{Q}}_{\mathbf{2}},{\mathit{Q}}_{\mathbf{3}},{\mathit{Q}}_{\mathbf{4}}\}$                $S{K}_2\leftarrow \{t_1,t_2,t_3,t_4\}$ 16: end function